# zbMATH — the first resource for mathematics

A congruence of generalized Bernoulli number for the character of the first kind. (English) Zbl 0982.11067
This paper deals with various congruences and their use concerning the generalized Bernoulli numbers $$B^n_\chi$$, where $$\chi$$ is a primitive Dirichlet character with conductor $$f$$. These numbers $$B^n_\chi$$ are considered to be elements of an algebraic closure $$\overline{\mathbb{Q}}_p$$ of the field $$\mathbb{Q}$$ of $$p$$-adic numbers ($$p$$ is an odd prime).
The first congruence (Proposition 1) gives a certain $$p$$-adic approximation of $$B^n_\chi$$ by means of a sum of the kind $$\sum_a\chi(a)a^n$$. This congruence is then used to derive a congruence of Ferrero-Greenberg (Proposition 5) concerning the derivative $$L_p'(0,\chi\omega)$$ at $$s=0$$ of the Kubota-Leopoldt $$p$$-adic $$L$$-function $$L_p(s,\chi\omega)$$, where $$\omega$$ is the character with conductor $$p$$ such that $$\omega(x)\equiv x\pmod p$$ for all $$p$$-adic integers $$x$$.
Using this congruence, the Ferrero-Greenberg formula [B. Ferrero and R. Greenberg, Invent. Math. 50, 91-102 (1978; Zbl 0441.12003)] is proved: if $$f$$ is prime to $$p$$, then $L_p'(0,\chi w)=(1-\chi(p))B^f_\chi\log f+\sum^{f-1}_{a=0}\chi(a)\log\Gamma\left(\frac af\right).$ $$\Gamma(x)$$ is the $$p$$-adic gamma function of Morita defined by $\Gamma(x)=\lim_{{n\to x,r}\atop {n>0}}(-1)^n\prod^{n-1}_{a=1,(a,p)=1}a$ for each $$p$$-adic integer $$x$$.

##### MSC:
 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 11B68 Bernoulli and Euler numbers and polynomials 11M38 Zeta and $$L$$-functions in characteristic $$p$$